Microsoft Word - 135-142 Mathematics | 135 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 The Comparison Between Standard Bayes Estimators of the Reliability Function of Exponential Distribution Mohammed Jamel Ali Moham198999@gmail.com Hazim Mansoor Gorgees Hazim5656@yahoo.com Adel Abdul Kadhim Hussein Adilabed57@yahoo.com Department, of Mathematics, College of Education for Pure Science Ibn Al-Haitham, University of Baghdad, Iraq, Baghdad Article history: Received 10 June 2018, Accepted 15 August 2018, Published December 2018"" Abstract In this paper, a Monte Carlo Simulation technique is used to compare the performance of the standard Bayes estimators of the reliability function of the one parameter exponential distribution .Three types of loss functions are adopted, namely, squared error loss function (SELF) ,Precautionary error loss function (PELF) and linear exponential error loss function (LINEX) with informative and non- informative prior .The criterion integrated mean square error (IMSE) is employed to assess the performance of such estimators . Key words: standard Bayes estimator, loss function, IMSE 1. Introduction The reliability theory is associated with random occurrence of undesirable events or failure during the life of a physical or biological system [1]. Reliability is a substantial feature of a system. Basic concepts related with reliability has been recognized for a number of years, however, it has got greatest importance during the past decennium as a consequence of the use of highly complex systems. In reliability theory, the exponential distribution has a distinctive role in life testing experiments. Historically, it was the first life time model for which statistical procedures were widely developed. Many researchers gave numerous results and generalized the exponential distribution as a life time distribution, particularly, in the field of industrial life testing. The exponential distribution is desirable because of its simplicity and its own features such as lacks memory and self-producing property. The probability density, cumulative distribution and reliability functions of one parameter exponential distribution are respectively defined as [2]: 𝑓 𝑑, Ι΅ ɡ𝑒 Ι΅ , 𝑑, Ι΅ 0 1 The cumulative distribution function is given by F t pr T t 1 e Ι΅ 2 R t 1 F t e Ι΅ 3 The one parameter exponential distribution is a member of exponential class of probability density functions which has the general form [3] f t, Ι΅ exp p Ι΅ k t s t q Ι΅ 4 then the exponential distribution the p.d.f. can be written as Mathematics | 136 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 f t, Ι΅ exp ɡ𝑑 𝑙𝑛ɡ Hence T βˆ‘ π‘˜ 𝑑 βˆ‘ 𝑑 is the complete sufficient statistic for Ι΅ and it can be easily shown that T follows Gamma distribution with parameters n and Ι΅ . 2. Standard Bayes Estimators The researchers employed three types of loss functions, namely, the squared error loss function (SELF), precautionary error loss function (PELF) and linear exponential error loss function(LINEX) . The Bayes estimator of the parameter Ι΅ is the value of Ι΅ that minimize the risk function R(Ι΅, Ι΅ where [4] 𝑅 Ι΅, Ι΅ 𝐸 𝐿 Ι΅, Ι΅ 𝐿 Ι΅, Ι΅ β„Ž ɡ׀𝑑 π‘‘πœƒ Ι΅ 5 In the case of squared error loss function we have 𝐿 πœƒ, πœƒ πœƒ πœƒ 6 Then, the risk function will be R Ι΅, Ι΅ Ι΅ Ι΅ Ι΅ h Ι΅Χ€t dΙ΅ = Ι΅ β„Ž ɡ׀𝑑 𝑑ɡ 2Ι΅ Ι΅β„Ž ɡ׀𝑑 𝑑ɡ Ι΅Ι΅ Ι΅ β„Ž ɡ׀𝑑 𝑑ɡ Ι΅ 𝑅 Ι΅, Ι΅ Ι΅ 2ɡ𝐸 ɡ׀𝑑 𝐸 Ι΅ 𝑑׀ Differentiating𝑅 Ι΅, Ι΅ with respect to Ι΅ and setting the resultant derivative equal to zero, we get: 2Ι΅ 2E ɡ׀𝑑 0 Solving for Ι΅ implies that Ι΅ E Ι΅Χ€t 7 ThePrecautionary error loss function is defined as [5]: L Ι΅, Ι΅ Ι΅ Ι΅ Ι΅ 8 If (PELF)is adopted, it can be in the same manner show that the Bayes estimator of Ι΅ is Ι΅ 𝐸 Ι΅ 𝑑׀ 9 Varian (1975) developed the following a symmetric linear exponential (LINEX) loss function 𝐿 βˆ† π‘’βˆ† βˆ† 1 10 Where βˆ† Ι΅ Ι΅ And when the ((LINEX) is adopted, similarity the Bayes estimator of Ι΅ is Ι΅ 𝑙𝑛 𝑒 Ι΅β„Ž ɡ׀𝑑 π‘‘πœƒ 11 Ι΅ 4. Posterior Density Based on Jeffrey's Prior Information Let us assume that Ι΅ has non informative prior density. Jeffrey's (1961) developed a general rule for obtaining the prior distribution of Ι΅ [6]. He established that the single unknown parameter Ι΅ which is regarded as a random variable follows such a distribution that is proportional to the square root of the fisher information on Ι΅, that is [5] 𝑔 Ι΅ 𝛼 𝐼 Ι΅ 12 That is Mathematics | 137 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 𝑔 Ι΅ 𝑐 𝐼 Ι΅ Where c is a constant of proportionality and I(Ι΅) represent fisher information defined as follows: 𝐼 Ι΅ 𝑛𝐸 πœ• 𝑙𝑛𝑓 𝑑, Ι΅ πœ•Ι΅ If g Ι΅ denote Jeffrey's prior information then 𝑔 Ι΅ 𝑐 𝑛𝐸 πœ• 𝑙𝑛𝑓 𝑑; Ι΅ πœ•Ι΅ 13 For the exponential distribution we have 𝑙𝑛𝑓 𝑑, Ι΅ 𝑙𝑛ɡ ɡ𝑑 ;Ι΅ Ι΅ Ι΅ - t Hence, βˆ‚ lnf t; Ι΅ βˆ‚Ι΅ 1 Ι΅ Substituting in Equation (13) it follows that 𝑔 Ι΅ 𝑐 Ι΅ βˆšπ‘› From Bayes theorem the posterior density function of Ι΅ denoted by β„Ž ɡ׀𝑑 can be derived as [4] β„Ž ɡ׀𝑑 , … , 𝑑 𝑔 Ι΅ 𝐿 Ι΅; 𝑑 , … , 𝑑 𝑔 Ι΅ 𝐿 Ι΅; 𝑑 , … , 𝑑 𝑑ɡ β„Ž ɡ׀𝑑 , … , 𝑑 Ι΅ 𝑒 Ι΅ Ι΅ 𝑒 Ι΅ 𝑑ɡ , T 𝑑 Hence, the posterior density function for Ι΅ based on Jeffery's prior information will be β„Ž ɡ׀𝑑 , … , 𝑑 𝑇 Ι΅ 𝑒 Ι΅ Π“ 𝑛 14 The posterior density in Equation (14) is defined identified as a density of the Gamma distribution, that is ɡ׀𝑑 , 𝑑 , … , 𝑑 ~ Gamma (n, ) with E(Ι΅) = and var(Ι΅) = that is Ι΅~Gamma (n , ) 4. Posterior Density Based on Gamma Prior Distribution Assuming that Ι΅ has informative prior as Gamma distribution which takes the following form: 𝑔 Ι΅ 𝛽 Ι΅ 𝑒 Ι΅ Π“ 𝛼 ; Ι΅ 0 , 𝛽 0 , 𝛼 0 15 Where Ξ±,Ξ² are the shape parameter and scale parameter respectively The posterior density function is β„Ž ɡ׀𝑑 𝑔 Ι΅ 𝐿 Ι΅; 𝑑 , … , 𝑑 𝑔 Ι΅ 𝐿 Ι΅; 𝑑 , … , 𝑑 𝑑ɡ Thus Mathematics | 138 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 β„Ž ɡ׀𝑑 𝑃 Ι΅ 𝑒 Ι΅ Π“ 𝛼 𝑛 16 Where P Ξ² T It can easily be noted that (ɡ׀𝑑 ~πΊπ‘Žπ‘šπ‘šπ‘Ž 𝛼 𝑛, with E(Ι΅) = ,Var(Ι΅) = ( ) . Bayes Estimator When (SELF)is Adopted 5 a: The case of Jeffrey's prior information. From Equation (7) we found that: Ι΅ E Ι΅Χ€t Ι΅β„Ž ɡ׀𝑑 π‘‘πœƒ Ι΅ n T 17 Similarly, the Bayes estimator of the reliability function can be obtained as follows: R t E R t tΧ€ R t h Ι΅Χ€t dΞΈ R t T T t 18 b: The case of Gamma prior distribution. In this case we have Ι΅ E h Ι΅Χ€t Ξ± n P 19 The estimator of the reliability function can be obtained as 𝑅 𝑑 E R t tΧ€ 𝑅 𝑑 𝑒 Ι΅ 𝑃 Ι΅ 𝑒 Ι΅ Π“ 𝛼 𝑛 π‘‘πœƒ Which, implies that 𝑅 𝑑 𝑅 𝑑 β„Ž ɡ׀𝑑 π‘‘πœƒ 𝑃 𝑃 𝑑 20 . Bayes Estimator When (PELF) is Adopted 6 a: The case of Jeffrey's prior information From Equation (9) we have Ι΅ 𝐸 Ι΅ π‘₯Χ€ The π‘Ÿ moment of ɡ׀𝑑 can be evaluated as follows: E(Ι΅ 𝑑׀ Ι΅ β„Ž ɡ׀𝑑 𝑑ɡ Hence, E( Ι΅ 𝑑׀ Π“ Π“ 21 When r=2, we get 𝐸 Ι΅ 𝑑׀ Π“ 𝑛 2 Π“ 𝑛 𝑇 𝑛 𝑛 1 𝑇 Hence, Mathematics | 139 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 Ι΅ 𝑛 𝑛 1 𝑇 22 Similarly, the Bayes estimators of the reliability function can be obtained as follows 𝑅 𝑑 E 𝑅 𝑑 𝑑׀ Now, we have to determine E[( 𝑅 𝑑 𝑑׀ E 𝑅 𝑑 𝑑׀ 𝑅 𝑑 β„Ž ɡ׀𝑑 𝑑ɡ = Hence, 𝑅 𝑑 𝑇 𝑇 2𝑑 23 b: The case of Gamma prior distribution From Equation (9) we have Ι΅ 𝐸 Ι΅ π‘₯Χ€ The π‘Ÿ moment of ɡ׀𝑑 can be evaluated as follows: 𝐸 Ι΅ 𝑑׀ Ι΅ β„Ž ɡ׀𝑑 𝑑ɡ 𝐸 Ι΅ 𝑑׀ Π“ 𝛼 𝑛 π‘Ÿ Π“ 𝛼 𝑛 𝑃 24 If r=2 then 𝐸 Ι΅ 𝑑׀ 𝛼 𝑛 𝛼 𝑛 1 𝑃 Hence, Ι΅ 𝛼 𝑛 𝛼 𝑛 1 𝑃 25 Similarly, the π‘Ÿ Bayes estimators of the reliability function can be obtained as follows 𝑅 𝑑 E 𝑅 𝑑 𝑑׀ Now, we have to determine E[( 𝑅 𝑑 𝑑׀ E 𝑅 𝑑 𝑑׀ 𝑅 𝑑 β„Ž ɡ׀𝑑 𝑑ɡ 𝑃 𝑃 2𝑑 𝑅 𝑑 𝑃 𝑃 2𝑑 26 7. Bayes Estimator When (LINEX) Is Adopted a: The case of Jeffrey's prior information From Equation (11) we have Ι΅ 𝑙𝑛 𝑒 Ι΅β„Ž ɡ׀𝑑 π‘‘πœƒ Ι΅ Mathematics | 140 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 Hence, Ι΅ 𝑙𝑛 𝑒 Ι΅β„Ž ɡ׀𝑑 𝑑ɡ By evaluating the integral we get Ι΅ 𝑙𝑛 𝑇 1 𝑇 27 The estimator of the reliability function can be obtained as 𝑅 𝑑 𝑒 Ι΅ 𝑅 𝑑 𝑇 1 𝑇 𝑑 28 b: The case of Gamma prior distribution From Equation (11) we have Ι΅ 𝑙𝑛 𝑒 Ι΅β„Ž ɡ׀𝑑 π‘‘πœƒ Ι΅ Hence, Ι΅ 𝑙𝑛 𝑒 Ι΅β„Ž ɡ׀𝑑 𝑑ɡ By evaluating the integral we get Ι΅ 𝑙𝑛 𝑃 1 𝑃 29 The estimator of the reliability function can be obtained as 𝑅 𝑑 𝑒 Ι΅ 𝑅 𝑑 𝑃 1 𝑃 𝑑 30 Simulation Study 8. The simulation study was conducted in order to compare the performance of the Bayesian estimators of the reliability function R(t)of one parameter exponential distribution. The integrated mean squared error (IMSE) as a criterion of comparison where 𝐼𝑀𝑆𝐸 𝑅 𝑑𝑖 βˆ‘ βˆ‘ 𝑅 𝑑 𝑅 𝑑 1 𝑛 𝑀𝑆𝐸 𝑅 𝑑 Where 𝑛 is the random limitsof 𝑑 , using t=(0.1,0.2,0.3,0.4, 0.5,0.6,0.7,0.8,0.9,1) L is the number of replications which we assumed that L=1000 in our study, 𝑅(𝑑 ) is the estimator of R(t) at the 𝐿 replication. The Bayesian estimators of R(t) are derived with respect to three loss function which are the square error loss function (SELF),precautionary error loss function (PSELF) and linear exponential error loss function(LINEX) , moreover, the informative and non-informative prior were postulated .The sample sizes n=10,50, 100 and 200 were chosen to represent small, moderate, large and very large sample sizes from the one parameter exponential distribution .The postulated values of the unique parameter Ι΅ were Ι΅=0.5,1.5 and the values of the parameters for Gamma prior were Ξ±=0.3,1 and Ξ²=1.2,3 . The results are presented in Tables below: Mathematics | 141 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 Table 1. (IMSE’s) values of the reliability function estimators by using Jeffrey's prior information at Ι΅=0.5 200 100 50 10 n Estimator 0.002635 0.004064 0.004635 0.000524 JSqu 0.002632 0.004041 0.004524 0.000484 JPre 0.002981 0.005875 0.010603 0.027725 JLIN Table 2. (IMSE) values of the reliability function estimators by using Jeffrey's prior information at Ι΅=1.5 200 100 50 10 n Estimator 0.001260 0.002309 0.003425 0.000963 JSqu 0.001259 0.002301 0.003376 0.000932 JPre 0.001279 0.002565 0.005140 0.019773 JLIN Table 3. (IMSE’s) of the reliability function estimators by using Gamma prior information at Ι΅=0.5 200 100 50 10 n Estimator 0.002618 0.003941 0.004250 0.000416 Ξ²=1.2 =0.3Ξ± GSqu 0.002543 0.003743 0.003806 0.000312 Ξ²=3 0.002609 0.003954 0.004457 0.000528 Ξ²=1.2 Ξ±=1 0.002556 0.003749 0.003899 0.000335 Ξ²=3 0.002614 0.003918 0.004178 0.000389 Ξ²=1.2 Ξ±=0.3 GPre 0.002538 0.003719 0.003739 0.000308 Ξ²=3 0.002605 0.003931 0.004384 0.000478 Ξ²=1.2 Ξ±=1 0.002551 0.003725 0.003832 0.000320 Ξ²=3 0.002981 0.005859 0.010439 0.027407 Ξ²=1.2 Ξ±=0.3 GLIN 0.002980 0.005817 0.010204 0.026386 Ξ²=3 0.002981 0.005859 0.010539 0.028030 Ξ²=1.2 Ξ±=1 0.002980 0.005830 0.010251 0.026935 Ξ²=3 Table 4. (IMSE’s) of the reliability function estimators by using Gamma prior information at Ι΅=1.5. 200 100 50 10 n Estimator 0.001250 0.002209 0.003000 0.000838 Ξ²=1.2 =0.3Ξ± GSqu 0.001227 0.002038 0.002462 0.001465 Ξ²=3 0.001250 0.002217 0.003055 0.000713 Ξ²=1.2 Ξ±=1 0.001227 0.002063 0.002497 0.001173 Ξ²=3 0.001249 0.002200 0.002949 0.000909 Ξ²=1.2 Ξ±=0.3 GPre 0.001226 0.002027 0.002410 0.001665 Ξ²=3 0.001249 0.002208 0.003004 0.000760 Ξ²=1.2 Ξ±=1 0.001225 0.002053 0.002446 0.001336 Ξ²=3 0.001279 0.002564 0.005123 0.018831 Ξ²=1.2 Ξ±=0.3 Mathematics | 142 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2006 Vol. 31 (3) 2018 0.001279 0.002564 0.005081 0.017556 Ξ²=3 GLIN 0.001279 0.002565 0.005127 0.019206 Ξ²=1.2 Ξ±=1 0.001279 0.002564 0.005087 0.017887 Ξ²=3 Simulation Results and Conclusions 9. From our simulation study, the following results are clear ο‚· From Table 1. and Table 2.: The Bayes estimator under ο‚· Precautionary error loss function with Jeffrey's prior is the best comparing to other estimators for all sample sizes. ο‚· From Table 3.: The Bayes estimator under precautionary error loss function with Gamma prior (Ξ±=0.3, Ξ²=3) is the best comparing to other estimators for all sample sizes. ο‚· From Table 4.: for (n=10) the performance of Bayes estimator under squared error loss function with Gamma prior (Ξ±=1, Ξ²=1.2) is the best, and for (n=50,100) the performance of Bayes estimator under precautionary error loss function with Gamma prior (Ξ±=0.3, Ξ²=3) is the best and for (n=200) the performance of Bayes estimator under precautionary error loss function with Gamma prior (Ξ±=1, Ξ²=3) is the best. ο‚· That is the performance of Bayes estimator under precautionary error loss function is superior to the performance of other estimators in almost cases that are studied in this paper, where the integrated mean squared error (IMSE) is employed as a criterion to assess the performance of such estimators. 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